IEEE TRANSACTIONS ONCOMMUNICATIONTECHNOLOGY VOL. 13, -NO.

’3 SEPTEMBER, 1965

total carried load = Pno,nb.na(n,+mnb<c+ms ACKNOWLEDGMENT

%.nb
The author wishes to exprcss his appreciation to A.
Descloux and M. Segal for helpful discussions; to Miss J.
Trask, who wrote the programs for generatingand solving
the state equations; and to Miss S. A. Switch who pro-
grammed the confirming simulations.
BIBLIOGRAPHY
COVERING RELATED
PROBLEMS
[ l ] R. M. Fortet, “Problems in the statistical theory of telephone
traffic,” in Digest of Technical Papers, Sixth Znternat’l Symp. on
Global Communications, p. 9, 1964.
[2] R. M. Fortet, and C. H. Grandjean, “Study of congestion in a
loss system,” presented at the 1964 Fourth Internat’l Teletraffic
Cong., London.
[3] J. Ott,erman, “Grade of service of direct traffic mixed with store-
and-forward traffic,” Bell Sys. Tech. J., vol. 41, pp. 1415-1437,
Jrlly 1962.

Error Probabilities Due to Atmospheric Noise and

Flat Fading in HF Ionospheric Communication

Abstract-Atmospheric noise and fading are major sources of (see [3], for example) have assunled Gaussian noise. The
error in the transmission of digital data via ionospheric reflection. present paper is written to fill the gap by calculating error
Using appropriate mathematical models of the fading and atmos-
rates due to both fading and atmosphericnoise. We assunle
pheric noise, this paper derives analytical expressions for the binary
error rate of FSK and PSK systems. It is demonstrated that the that the fading is slow, nonfrequency selective (i.e., flat-
error rate depends on the atmospheric noise in the various diversity flatin the terminology of [4]), and characterizableby
receivers only through a single composite noise variable equal to the means of a complex valued Gaussian process (as in [4]).
s u m of the detected noise powers of the outputs of identical diversity The effects of time and frequency selective fading have
receiver filters. At large SNR’s and Lth order diversity the error
been studied elsewhere [4]-[9]. Our assumptions with re-
rate is shown to be proportional to the reciprocal of the SNR raised
to the Lth power. Simple expressions are derived showing the system gard to the properties of atmospheric noise are detailed
degradation resulting from the presence of atmospheric rather than in Section IT following. The mathematicaloperations
Gaussian noise. A comparisonof theoretical and measured error characterizing basic FSK and PSI< matched filter receivers
rates for the AN/FGC-29 and the AN/FGC-54 shows remarkably employing diversity combining are defined in Section 111.
good agreement. Section IV derives the error probability forarbitrary inter-
ference assuming matched filker FSK and PSI< systenls
I. INTRODUCTION using maximal ratio diversity combining in the presence
of flat (complex) Gaussian fading. Analyticaland numerical
T HE PERFORMANCE of a digital data modem over
a n ionospheric HF link is limited by both fading and
additive noise, the latterbeing primarily atmosphericnoise.
results on error probabilities with atmospheric
presentedinSection
noise are
V. Section VI summarizes some
Previouscalculations of error rates due to atmospheric important conclusions.
noise [l], [ 2 ]have assumed a nonfading signal, while those IT. ATMOSPHERIC
NOISE
CHARACTERISTICS
calculations of error rate including both noise and fading
The term “atmospheric noise” has been employed with
Manuscript received May 24, 1965. Presented as paper CP65-533 somewhat different meanings in the literature depending
at the 1965 IEEE CommunicationsConvention,Boulder, Colo. upon the point in the receiver a t which it is measured.
The work reported in this paper was supported in part by the U.S.
Army Electronics Laboratory, Ft. Monmouth, N. J., under Contract However, no ambiguity exists as to the source of atmos-
DA-28-043 AMC-00038 (E). pheric noise, namely lightningdischarges. These discharges
The author is with SIGNATRON, Inc., Lexington, Mass. He was
formerly with ADCOM, Inc. occur randomlyintime and geographical location and
266
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 PROBABILITIE3
ERROR
BELLO: I N HF IONOSPHERIC
COMMUNICATIONS 267

propagate over long distances via ionospheric reflection. though NBS gives curves which allow one to take account
The collective nature of these discharges is such as to of a change in bandwidth on the envelope probability dis-
produce random pulses of electromagnetic field a t a given tribution, no curves are given to allow one to determine
receiving antenna.These pulses are of the “low pass” the effect of a change in filter shape. We shall make the
type, i.e., their perceptiblespectrumextendsfromvery assumption that the NBS envelope distributioncurves
low frequencies up to around30 hilc/s. Thus it is not mean- apply to our receiver filters also for lack of the correct
ingful to talk about either the phase or envelope of the measured distributions. However, this assumption is not
received atmosphericvoltageinahypothetical infinite likely to makeour final answers less useful for performance
bandwidth receiving ant.enna. However, antennaband- prediction than if we had the correct measured distribu-
widths are finite, and, more to the point, the filters used to tions, since the random variationsof measured distributions
detect HF digital signals are narrow-band filters centered a t a given geographic location and in a given time block
in the HF band. Thus in determining error rates for digital will probably introduce sufficient unavoidable prediction
systems used over the HF ionospheric medium, one must error to obscure the foregoing approximation error.
deal with an interfering noise at thedetector output which We now present certain properties of the lognormal dis-
is the result of cxciting a narrow-band filter with randomly tribution needed for the subsequent development. If I is a
occurring “low pass” pulses havingrandomamplitudes lognormally distributed random variable, then it may be
and shapes. represented as
Measurements of atmospheric noise have dealt almost
1 = eg (1)
exclusively with the envelope at the output of a narrow-
band filter. An extensive series of such measurements where g is a normally distributed random variable. The
carried out on a worldwide basis and over a wide fre- probability density function of 1, W1(l)is related to that
quencyrange has been accomplished by the National of g, W,(g) by theprobability equivalence relation
Bureau of Standards (101. Thesemeasurements include
both noise power and probability distributions. Although WL(W = W,(g)&7. (2)
measurements were performed for a 200 c/q bandwidth, Since
correction curves are given to allow conversion of the
results to other bandwidths. I t has been found [lo] that
the probability distribution of the envelope is close to the
Rayleigh distribution only for the small-noise high-proba-
bility levels. However, in the high-level low-probability where p and u are the mean and standard deviation of g,
region there is marked departure from Rayleigh, the meas-
ureddistributionhavingamuch longer “tail.”Sucha
distribution might have been expected a priori, the small
noise levels being caused by the overlapping of a large The sthmoment of 1 is given by
number of low level atmospheric discharges and the large -1 = -e@ = ePse+82/2
noise levels by much fewer distinct atmospheric “spikes” (5)
extending abovethe ambient noise level. where the overline denotes an ensemble average, and the
It has been found [11], [la] that the peak values of the averageover g may be recognized as the characteristic
largeatmospheric noise spikes haveaprobability dis- function of g.
tribution which conforms rather well with the lognormal I n practice, it is the probability distribution function
distribution. This fact, no doubt, is the reason why the rather than the probability density function of the noise
tail of the lognormal distributionfits closely with the envelope that is measured. Thus if e denotes the envelope,
tail of the measured envelope probability distributions, as what is measured,in effect, is the probability that the
pointed out in [13]. For signal-to-noise ratios of practical envelope (normalized with respect to the rnls value) ex-
interest it is primarily the spikes of noise that cause errors1 ceeds a threshold r, i.e.,
and thusit is primarily the tail of the noise probability dis-
tribution that is of interest. Consequently, as far as error Pr[+ > r].
rate computation is concerned, it appears reasonable to
assume that the envelope detected noise distribution in
the receivers to be analyzed can be approximated by a In thecase of a lognormally distributed random variable
lognormal distribution, which has the same “tail” as the 1,
true measured probabilitydistribution.Unfortunately,
in the receivers to be analyzed, the narrow-band filters do
not have the same transfer function or bandwidth as the
200 c/s filter used for noise measurements by NBS. Al- where the error function
n ”“,

1This statementis not likely to be true for nondiversity operation,

as discussed at the end of Section IV.

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268 IEEEON
COMMUNICATION
TRANSACTIONS TECHNOLOGY SEPTEMBER

If the normalized 1 and r are expressed in decibels, equal to u if the distribution function is lognormal. As-
suming that the lognormal character of t,he measured dis-
1
L = 20 log10 tribution has been reached by the 2.28 percent level, it is
~

q then easy to find the value of B for the approximating

lognormal distribution. If it is felt thatthe lognormal
character has not been reached by the 2.28 percent level,
then the distribution function
becomes then it is easy to find other pairs of percentiles for which
(R/8.686 + "')I. (11)
the difference in levels is simply related to B.
Using the foregoing approach and studying in [lo] the
values of Td' measured in the HF band, it is found (with
appropriate adjustment of these values for a filter band-
In the extensive measurements of atmospheric noise
width of 100 c/s rather than 200 c/s) that B lies in the
envelope taken by NBS, it was found that two parameters
range
were sufficient to characterize the probability distributions,
namely the rms value and the ratio of the rms value to 2dB<B<GdB (20)
the mean value. Lognormal noise may also be character-
for the HF band. If narrower band filters are of interest,
ized by these two parameters. Letting the former parameter
the range of B values mill decrease. Values of filter band-
be denoted by a and the latter by b we find from ( 5 ) that
width much higher than 100 c/s are not likely to be of
-
a = diz = eweo2 (12)
interest for error rate
analyses, since such filter bandwidths
imply modems operating with subchannel data rates high
enough to cause severe nlultipath distortion.
OPERATIONS
111. RECEIVER
Using decibel measures of a and b, Simplified block diagrams of the basic FSK and PSI<
receiver configurations are shown in Fig. 1. Two types of
A = 20 log10 a (14)
PSK systems are illustrated. The PS-PSI< system uses the
B = 20 log10 b (15 ) previous signaling element as a phase reference, while the
PT-PSI< system uses apilottone as a phase reference.
we find that Although demodulation of only the in-phase modulation
component of the PSI< system is shown, there will also be
A = (p + a') 8.686 (16)
a demodulator for the quadrature modulation component
B = u2 4.343 (17) when the system is running quaternary, i.e., when both
in-phase and quadrature modulation is used.
and thus Note that square law envelope detection is employed in
theFSK system The diversity combiners simply add
A - 2B corresponding components of the variousdiversity re-
p=-
8.686 ceivers. Aftersampling the diversity combined output,
the result is compared to a zero threshold.
We shall assume statistically stationary conditions and
evaluate theerror probability on anensemble basis, assum-
We note from (11) and (19) that the (normalized) dis- ing that a received signaling element occupies the time
tribution function depends upon the single parameter B. interval 0 < t < T . It will be shown that the sampled out-
The (normalized) distributionfunctions of atmospheric puts of each of the systems can be represented as a quad-
noise envelope measured by NBS depend upon the single ratic form of the type
parameter V d , which is entirelyanalogous to B, i.e., it L L
is the ratio of the rms to themean noise envelope expressed QL = c 1% +
p=l
tPI' c
-p=l l'lP1' (21)
in decibels. As discussed previously, we desire to approxi-
mate the measured distribution function (called APD for where L is the order of diversity, Gp is a complex signal
amplitudeprobabilitydistribution)by the lognormal term due to the pth diversityreceiver, and F,, 71,are com-
distribution function (11). Our problem is then, given a plex noise terms due to the pth diversity
receiver.
value of V d , find the value of B for which the distribution In thecase of FSK the complex envelope of the received
function (11) forlarge R coincides with the measured signal for a unity gain channel and no additive noise may
distribution functioncorresponding to thevalue of V d . be taken as
A simple method for determiningu and thusB has been
suggested by Beckmann([13],p. 735). He points out
that the decibel difference between the 2.28 percent and
0.135 percent levels (i.e., values of L for which P [ L > where is +1 for mark and - 1 for space,j,, is the carrier
cyQ

R ] = 2.28 X and 0.135 X divided by 8.686 is frequency, &(t) is the rectangular pulse,

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1965 B E L L O : ERROR P R O B A B I L I T I E S I N HF I O N O S P H E R I C C O M M U N I C A T I O N S 269

r Mork FromOtherReceivers
FSK
Ftlter

Input
-i 1 1
f

Sub.
7

Threshold
Sample and
Dlversity output
1
Spoce
FI Iter

PS-PSK ' Matched

Filter I
- 7
FromOtherReceivers
I

Detector
Comparison

Filter
with
Delay

FromOther Receivers
PT-PSK - 1 1

--
Product Zero
Input + --+
Threshold Outpu1
Detector
Comparison

Fig. 1. Simplified block diagram of FSK, PS-PSK, andPT-PSK receivers.

Q(t) = {Ol ;;Ot <<Ot ,<t >TT The approximation in (27) comes from the slow fading
hypothesis.Uponsumming the sampled outputs for all
diversity channels and using asummation index p , we
T is the pulse duration, and l / T is the mark-space fre- arrive a t (21).
quencyseparation. For the PS-PSI< system the complex envelope of the
The complex envelope of the received waveform for a received signal for a unity gain channel and no additive
particular diversity channel, including frequency-flat fad- noise is given by
ing and additive noise, is given by
m

~ ( t )=
-m
C Q(t - Q T ) an (30)
where n(t) is the complex envelope representation of the
where a y is a complex number of unity magnitude defining
additive noise and g(t) is a complex valued Gaussianprocess
the phase of the qth data pulse. Digitalinformation is
characterizing the fading.
contained in the product a*q--laqsince the q - l t h pulse
The sanlpled output of the receiver is given by
is used as a reference for the yth pulse. For quaternary
nlodulation

wherein it has been assumed that the mark and space

filters are matched filters. From symmetry considerations
it is sufficient to consider the evaluation of error probability while for binarynlodulation
for the case in which a nlark hasbeen received in the inter-
val 0 < t < T . Then using (23) and (24), in ( 2 5 ) , we find
that The sanlpled output of the in-phase channel at the time
Q = IG + $I2
- IT[' (26) t = T is given (interms of operationson complex en-
velopes) by
where
PT
G = J, g(t)dt = g(0)T (27)

s T (where -~'gis a convenient scale factor that does not.

n(t) dt change probabilities since q is compared with zero) while
= 0
the sampled output of the quadrature channel is given by
T
,jd/ T the imaginary part of the bracketed term in (33). From
n = n(t) dt.
symmet,ry considerations it is sufficient to consider the

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270 TRANSACTlONS I E E E SEPTEMBERTECHNOLOGY
ON COMMUNICATION

evaluation of error probability for the in-phase channel s(t) in (24) takes theform shown in (30) with
alone, assuming that

for the quaternarycase and

Use of (30), (34), and (24) then shows that a@= * l (45)

q = Re{ d/Z[a-l*G* + 7*][aoG+ e ] } for the binary case.

Upon carrying out the integrations in(43) and using the
where we have used the approximations slow fading assumption, we find that for the quaternary

lT g(t) d f = s_”, g ( t ) dl = g(0) T = G

case

p = R e ( fiG*[aoG + e]} (46)

and thedefinitions where 0 is given by (38). Since (46) is a special case of

(30), it is readily seen that q in (41) can be expressed in the
form of (21) with F, 9 given by

e = LT n(t) dt.
It is assumed in the foregoing that the error probability
(47)

Upon multiplying out the terms in (35) and rearranging of interest can be confined by symmetry considerations to
terms, one finds that q in (35) can be made to take the the case in which
form (26), where 5, r] take different values, namely,

5=
a*oy + a-l*e (39)
4 For the binary PT-PSK case, the variables 5, 7 can be
chosen as

(49)
It is readily determined that in the case of a binary PS-
PSK system the sampled output can also be represented
We have thusarrived at theconclusion that each of the
in the generic form (21), but where E, 7 take thenew values
systems of Fig. 1has a diversity-combined sampled output
expressible as the quadraticform (21) withappropriate
interpretations of t, qr for each system as outlined in the
foregoing. Note that we have defined transmitted signals
such that the sampled output at t = T will be detected in
error if qL < 0.
Before proceeding to thederivation of error probabilities
where a-l*a,, is given by (32). Thus, for both binary and we shall briefly discuss the characterof the signal and noise
quaternary PS-PSK, the diversitycombined output can be terms in (26) [and thereby in (al)]. The signal term G is a
made to take the form (21). complex valued normally distributed variable which, be-
Turning now to the PT-PSI< systems we note that the cause of the slow fading assumption, may be taken as the
complex envelope of the received signal has the same form sameforallsystems. For simplicity, andwithout loss
as for the PS-PSI< system. However, instead of the previous in generality of our final results, we shall normalize G so
signaling element being the phase reference, atrans- that
mitted pilot tone provides the phase reference. Neglecting
noise on the received pilottone, its received complex ‘ / z jG12= 1 (50)
envelope will be proportional to g ( t ) , and for convenience
where the overline indicates an ensemble average.
(with no loss in generality of our results) we select this
The signal-to-noise ratio for each system will be defined
proportionality constant equal to T . Thus for the sampled
as theratio of the received signal power to the noise power
output of the in-phase channel, we find (in terms of com-
inabandwidth 1/T. Weshall assume the atmospheric
plex envelopes) that
noise to bespectrallyflatover the matched filter pass

q = Re{ z/z lT Tg*(t) s(t) dt}

band. Then we may model n(t) as a white noise with spec-
(43) tral density No.
Since the real noise power is one-half the “power” in the
where y(t) is,given by(24). In thecase of PT-PSI<systems, complex representation, the input signal-to-noise ratio p

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1965 BELLO: ERROR PROBABILITIES I N HF I O N O S P H ~ I C ^ C O M ~ ~ ~ ~ ~ ~ : L I O N S . i 271

is given by where we have defined

L
u = lI;P = c
p=l
15P12 (59)
L
v = In12 = [qp12. (60)
9-1

Q ( a ,p) is Marcum’s Q function defined by

IV. ERROR PROBABILITY
EXPRESSION
FOR ARBITRARY
ADDITIVENOISE
For each of the digital modems, we have defined the
Q ( a ,p) = Lm IL: exp [- (x2 + a 2 ) / 2 ] I o ( a xdx) (61)

transmitted signal in 0 < t < T such that an error will and I,( ) is the modified Bessel function of the first kind
occur if the sampled diversity combined output is negative, and order r.
].e., We note the interesting fact that the conditional error
.I 1
probability depends onthe additive noise only through the
two generalized noise powers U and V , i.e.,

PL = fL(U, VI. (62)

Consider now the evaluation of the conditional prob-
ability of error given a fixed set of sampled noises tl, &, . . . Thus to find the unconditioned error probability p,, it is
tp,ql, q2, . . . q p , i.e., considcr necessary to average only over the pair of variables U , V ,
rather than the 2L random variables [I, . . . EL, ql, . . . .7,
PL = Pr
L

[ p = l
1,;’ <
/Go 4~-
P = l
~ ~ , ~ 2 iI [ o ,
I
’q0 p = 1, 2 , . . . L
I
(53)
. Let the joint probability density function of U , V be de-
noted by W,(U, V ) .Then

If we mentally reverse the role of noise and signal and

PL = SS W L ( ~V,) . f ~ (Vu) , dV- (63)
regard {Ev.;p = 1, 2, . . . L } , ( q p ;p = 1, 2, . . . L } as com- Now a close examination of theinput .anddetected
ponents of (complex) signal vectors noises reveals a symmetry in the statistics of the noises
&,,q p which leads us to conclude that
f = (41, E2 ... tLI (54)
WdU, V ) = W,(V, U ) . (64)
n = (VI, 72, . . . ml (55)
This being the case we can express p , in an alternateform
and { G p ;p = 1, 2, . . . L ) as components of a complex involving only the symmetric part of f L ( U , V ) ,i.e.,
noise vector
G = {GI, Gz, . . ., G L ) (56)
then we can identify the conditional error probability as
By making use of the identity (see [14], page 153, 3.15)
the probability that thelength of a signal plusnoise vector
is less than a threshold given by the length of another Q(v‘U, .\/TI + Q(.\//v, .\/u) = 1 + e--(u+v)’2
signal vector, i.e.,
Io(V‘/uv) (GG)

PL = PT[IG + tl < jnl I;, .I. (57)

we find that thesymmetric part of j ( U , V ) is given by
1
Assunling independentlyfluctuatingdiversity channels,
SL(U, V)= 2 [.fL(Ul V) + fL(V1 Wl
the “noise” vector G has Cartcsian components that are
mutually independent complex valued Gaussian variables.
The foregoing mathematical problem arises in the com-
putation of radar detection probabilities for the multiple
observation case. From the results of Helstrom [14] on this
problem2 we readily find that
where it should be noted that
I-T(x) = Ir(z). (68)
Thus by using the symmetric property of W,(U, V )we
have eliminated the Q function from the double integral
defining p,.
To evaluate the error probability p , it is necessary to
2 See Helstrom [14], section VI.2, eq. 2.19, withf = 1, S = If, L =
determine the joint probability density function of U and
:n’yields 1 -PL. V , W(U, V ) .In thecase of the PT-PSI< systems it is seen

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 273
SEPTEMBERTECHNOLOGY
COMMUNICATION ON IEEE TRSNSACTIONS

from (47) and (49) that widths less than 1/75 second we shall assume U = V in
our calculations for the FSIi system. It may be assumed
that the statistics of the narrow-band filter output do not
change significantly if the filter center frequencyis changed
over frequencies a t least as large as the mark-space fre-
quency separation of an HF-FSII system. Thus [see (28),
(29) ] we may take
when n, is the additive noise in the pth diversity channel
and thus
u = v. (70)
for the FSK system. A comparison of (69) and (74) reveals
Consequently for the PT-PSI< system that when our assumption U = V is valid for the PSI<
W,(U, V ) = WL(U)S(U - V ) (71) system, the quaternary PT-PSI< system is exactly 3 dB
better than the FSII system and the binary PT-PSI<system
where W L ( U )is the density function of U . Then defining is exactly 6 dB better than the FSK system.
We now turn ourattentiontothe PS-PSI< systems.
Here we may also argue4 on the basis of assumption 1) and
assumption 2) modified so that T is replaced by 2T, that
we find that p , is given by U = V . Thus if theseassumptions are true, an error is
Prn caused primarily by an “impulse” occurring in only one
of two adjacent signaling elements. Then, neglecting the
low level atmospherics, either y, or e, [defined in (37) and
I t will now be argued that in the case of the FSKsystem (38)], but not, both, is nonzero. As a result it is readily seen
one may also assunle U = V and still arrive a t useful re- from (39)-(42) that
sults. The argumentrunsas follows. Errors caused by
atmospheric noise3 are due prinlarily to the large occa- I
sional spikes rather than the small frequent noise pulses. c/
According to Horner and Hanvood[l] the effect of a large IEPl = IVpl = (75)
spike on a filter output is to produce a response essentially
identical to that of an impulse at the input. This effect
may be justified from the simple physical argument that
cI
where the constant c is l / d 2 for the quaternary case a.nd
the mrrow-band filter intercepts energy from only a small
1/2 for the binary case.
portion of the “tail” of the spectrum of an atmospheric Since 8, and y , have the same statistics, and since the
pulse. If the spectrum of the pulse is essentially constant
events e, fO, y p #O are mutually exclusive, the error
over the filter pass band,the filter cannotdistinguish
probability for the PS-PSI< can be computed as twice the
the input from an impulse input. Now in the case of a
error probability for the case
single impulse intercepted by the PSI< filters it is readily
verified that Itp] = Iqp/ and thus U = TT. Thus in order to
assume U = Tr for t,he FSK receiver and not nlaterially (‘76)
affect the accuracy of our calculated error probability, the
following conditions must pertain. Since these values of Itp[,l~,l are the same as for the PT-
1) Errors are caused primarily by the occasional large PSI< system, we conclude that, with assumptions 1) and
spikes of atmospheric noise. “modified” a), the errorprobability for binaryand
2) The frequency of occurrence of these spikes is small quaternary PS-PSI< systems is twice that of the binary and
enough so that in the duration of a signaling element ( T ) quaternary PT-PSI< systems, respectively. This result is
the probability of more than one impulseoccurring is act.ually obvious by inspection since impulses, if they cause
small. errors, cause double errors in PS-PSI< systems but only
An examination of experimental data [ll]indicates that single errors in PT-PSK systems.
such a pair of assumptions are valid for sufficiently small To summarize, if
values of signaling elementduration. The data are not
PLPT- 2psK = fL(P) (’77)
extensive enough to allow a definite choice of the mini-
mum value of T for which 2 ) is valid, but it appears safe denotes the error probability of the binary PT-PSI< sys-
to say that values of T corresponding to the typicalsignal- tem for Lth order diversitya t a signal-to-noise ratio p,
ing element duration of a teletype system,ie., 1/75 second
still allow the use of assumptions 1) and 2 ) . Thus for pulse PLPT- 4psK = fL(P/2) (’78)
pLFSK = fL(P/4) (792
3 This statement does not apply for nondiversity operation. See
discussion a t end of this sectlion. Again, nondiversity operation requires special attention

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1965 BEL1.0:PROBABILITIES
ERROR
IN IIF IONOSPHERIC
COMMUNICATIONS 273

PS-PPSK =
PL 2"fL ( P ) (80) averages in (87) are the same for all diversity brauches.
PS--4PSIi - Now since n(t)is a whitenoise of power density N o ,
PL - 2fL(P/2) (81)
T
-
where the notation p L L signifies the binaryerrorprob- n*(t)n(s)df.d s = NOT (90)
ability of system 12: with Lth order diversity, and 4PSK,
2PSK denote quaternary and binaryPSI< systems, respec- and we find that
tively. It should benoted that some of the transmitter -

power must be devoted to the pilot tone i n the case,of the u = L/2p
PT-PSI\; system. Thus fora given transmitter power, where p is the input SNR [see (51)].
the PT-PSI< system will suffer areductionin received An asymptotic series expansion of fL(p) in powers of l / p
SNR relative to theFSK and PS-PSI< systems. can be obtained by expressing GL(U) in (85) or G L ( U ) in
The function f L ( p ) is given by the integral in (73) with (73) as a Taylor series and then integrating term by term.
W , ( U ) given by the probability density function of From the expansions

(92)

where the random functiotlal

it is readily determined that

is just the envelope of atmospheric noise observed at. time
T at the output of a narrow-band filter whose impulse
response is a rectangular RF pulse of duration T .
The stutistics of U have never been measured. When any by integration that
they are measured they will be placed ; I I the form of an
APD, say,

Fdu) = P r [ i >u] = 1 WLo(U)

dU (84) Because of the alternation insigns in the series (94) and
(95), we have thefollowing upper andlower bounds:
where WLo(U)
to w L ( U ) by
is a normalized density functionof U related
(3
L-I 1
r1- 2
2(L - 1)!L
U2L+l
(T)]
< GiL(U)< (F>"-'
WLO(U) = UW,(UV). (85) 1
2(L - l)! (96)
From the point of view of calculating error probabilities
using measured APD's, it is desirable to have an expres-
sion forfL(p) which uses FL0(u) directly. Suchan expression
is readily obtained by integrating (73) by parts. Theresult
is Use of (96) in (86) and (97) in (73) leads to thefollowing
bounds onp L :

cP2, ( l _ : ) < p L < - CL

PL

where
where

and

and
If we assume that the noise statistics are the same for umWLO(u)du
= m ~P-~F~o(u)du
(101)
each diversity branch, then
is the mth moment of the normalized noise distribution
WLo(u). One may regardWLo(u) as thedensity functionof a
random variableu obtained from U by normalizing to unit
where the p subscript has been dropped since the required average value. The bounds in (98) will be useful only if

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274
SEPTEMBERTECHNOLOGY
COMMUNICATION ON IEEE TRANSdCTIONS

theyare close together.Thus SNR’s should belarge recterrorprobabilityforGaussianadditive noise and

enough so that SNR’s above 10 dB, it maybe concluded that atmospheric
noise willbe less harmful thanGaussian noise in thecase of
P >> DL (102) nondiversity operation. However, as will become clear in
if the approximation implied by (98) is to be close. It is Section Ti, atmospheric noise is more harmful with diversity
important to note that, for Lth order diversity the error reception. This difference in behavior between diversity
probability always falls off as l/pL, regardless of the addi- and nondiversity operation can be explained in the follow-
tive noise probability distributions. For sufficiently large ing way. Errors are caused not only by large noise spikes
SNR’s the relative errorprobability for two different addi- but also by the signal fading into the low-level noise even
tive noises is just given by the ratio of the moments 2 for when the large spikes are not present. Diversity operation
the two different additive noises, so that makes a radical change in the fading statistics, causing the
contribution to
the error
probability
due
to noise
p L withatmospheric noise
_ _-- -. with atmospheric noise
spikes to outweigh thatduetothe fading. I n nondi-
p , with Gaussiannoise uLwith Gaussian noise; versity operation, however, the reverse is true. As a con-
large SNR. (103) sequence, the large noise spikes occurring duringfades
represent “wasted” or ineffectual noise power. It may be
Inthe case of independent identically
distributed concluded that, in the case of nondiversity operation, cal-
Gaussian noise in each. diversity channel, U is distributed culations of error probability a t HF based upon Gaussian
as a chi-squared distribution with a2L degrees of f r e e d ~ r n . ~ratherthan atmospheric noisewill providea conserva-
For this distribution tivesystem design. Since we are primarilyinterestfed
- 1 (2L - l)! here in situations wherein the atmospheric noise is more
uL -- -
(Gaussianadditive noise) (104) harmful than Gaussian noise we shall not consider nondi-
LL (L - l)!
versity operation further.
so that for the Gaussian noise case6
v. ERRORPROBA4BILITIESDUETO ATMOSPHERIC
NOISE

(Gaussianadditive noise) (105) In order to use (73) to evaluate the desired error prob-
abilities, it is necessary to determine the probability dis-
where (r)is the usual binomial coefficient notation tribution of U in (82). This random variable is the sum of
squared envelopes of filtered atmospheric noise for identical
nz !
(3 = n!(vt - n)!
filters in different diversity branches. For convenience we
repeat (82) :
It follows from the foregoing that, in thecase of general
additive noise and abinaryPT-PSKsystem,the error
u = 41- p=l
=
1,2. (108)
probability a t sufficiently large SNR’s [see (loa)] is given
11,we shall assumeI, to be
As discussed in detail in Section
by
lognormally distributedwithdensityfunction(4)and
APD (distribution function) given by (7) or (11). As in (5)
we may associate with I, a normally distributed variable
g, such that
Asymptotic error probability expressions for the other
systems may be obtained via (78)-(81). Although (79)- 1, = eyp (109)
(81) are approximations, it may be shown that they yield where g p has mean p and standarddeviation U. Since
exact expressions for the asymptotic error probabilities.
To demonstrate this fact it is necessary to use (65) and 1,2 == 289, (110)
expand gL ( U , V ) in a double Taylor series in U and V .
lP2 is also a lognormally distributed variable with a n asso-
Upon defining normalized U , V variables and integrating
ciated Gaussian variable of mean 2p and standard devia-
term by term, an asymptoticseries in reciprocal powers of
tion 2a.
p may be obtained.
It follows that U is given by the sum of a set of possibly
An examination of (107) reveals the interesting fact that
dependent lognormally distributed randomvariables.
the error probability for large SNR and nondiversity opera-
Since the joint statistics of atmospheric noise on different
tion isindependent of thenoisestatistics. Since the ex-
diversitybranches have never been measured, we are
pression (107) is an upper bound, and since it may readily
handicapped in our efforts to determine the probability
be verified7 that this upperbound is quite close to the cor-
density function for U . The approach followed here is to
See Cramer [15], pp. 233-234. make calculations for the two extreme cases of complete
6 Pierce [3] has previously derived this asymptotic expression. dependence and complete independence of the 1,’s. One
l ? * 7
may expect thattheactual errorprobability will lie
7 For Gaussian noise the exact valueof p~ is
between these two extremes and that the error probability

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1965 INPROBABILITIES
ERROR BELLO: COMMUNICATIONS
HF IONOSPHERIC 275

with independent noises will be a lower bound. The latter In the case of dependent noises the SNR degradation
expectation is a consequence of the observation that with Sdeg resulting from atnlospheric noise as opposed to Gaus-
diversity operation it is the tail of the noise distribution sian noise a t large SNR's is readily obtained by a com-
which causes most errors, and that a sum of n independent parison of the asymptoticerrorprobability expressions
identically distributed noise has a shorter "tail" than n (105) and (113). This degradation is
times any one of the noises.
Before presenting error probability curves we shall dis-
cuss the asynlptotic error probability expressions for the
independent anddependent noise cases just discussed. where the parameter B is related to u2 by (19). In the case
The generic form for this high SNR error probability is of 2nd- and 4th-order diversity, this degradation becomes
given
- by (107) in terms of a single unknown, the moment (2B -0.88) dB and (6B -4.3) dB. When the noises are
uL.This nloment is the Lthmoment of the variable U nor- independent the degradation is smaller by an amount less
malized to unit mean.Thus than (116).
-
UL = [$-p =
1
[L
L
1 I,z,p].
L
(111)
The exact error probability is given by the integral in
(73) which may be expressed as the average

PL = GL(W. (118)
In the case of identical detected noises on the diversity
branches For the dependent noise case (Il = l2 = 1) and the PT-
- 2PSIi system
UL = e 2&[L*-L1 (112)
and thusfrom (98), (loo), and (107) we see that

where 1 is the common detected noise and g is the asso-

ciatednormallydistributedvariable of mean p and
where the d subscript denotes that f d L ( p ) applies to thede- variance u2. We may express g in terms of a zero mean
pendent noise case. unit variance random variable x as follows:
When the detected noises are independent, g = ux + p. (120)
- 1
uL =
L

p1=l
c
pr=l
L
...
L

PL=l
~~

lD,21P22.. . lp,2 Using (119) and(l20),the average in (118) canbe ex-

pressed as an average over x. Thus,

= [h]"[LF + L2(L - l)F=P + ...I PLPT-2PSK

=
L-1
e?a*[L*-L]
. (114) -~
-
1
2/2R s-mm
Since the ratio of asymptotic error probabilities for two From (12)-(19) and (90) we find that
different additive noises is just the ratio of the correspond-
ing u" moments, we conclude from (113) and (114), and B
&2 - --
the readily deduced fact t,hat is largest when all the ZP2 4.343 (122)
are equal, that S - 3 + 2 B-
K = (123)
8.686
(115)
where S is the SNR in decibels. With the aid of a digital
computer, the integral (121) has been evaluated as a
where the i subscript on f i L ( p ) denotes the independent
function of X for L = 2 and B = 2, 4, 6. From these
noise case. Since the asymptoticerror probability decreases
curves for the PT-BPSK system shown in Fig. 2 we can
as the reciprocal of the SNR raised to the Lth power, it
generate corresponding curves for the other four systenls
maybe seen from (115) that the reductionin SNR (to
via (78)-(81). Thus, if we let
achieve a given error probability) caused by independent
rather than dependent noises is less thau PT-2FSK -
Pz - FL(S) (124)

10 (y) logloL dB.

then
P2
PT-4PSK -
- FL(S - 3) (125)
For 2nd- and 4th-order diversity this is just 1.5 and 4.5 pzFSg =
FL(X - 6) (126)
dB, respectively. It should be noted from (111) and (114) PS-ZPSK -
that (115) and thus (116)is true for any additive noise Pz - 2FL(S) (127)
PS-4PSK =
statistics. Pz 2FL(S - 3). (128)

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 IEEE TRANSACTIONS ON COMMUNICATlON TECHNOLOGY
SEPTEMBER

Fig. 2. Generic error probability curves for flat fading and atmos-
pheric noise. Dual diversity. Detected noises on
receivers assumed identical.
diversity

For comparison purposes we have plotted also the error

probability for the case where independent Gaussian noise
is the only interference. Note that the predicted degrada-
I20

S in de +
25

pheric noise. Dual diversity. Detected noise on diversit,?-

receivers assumed independent.

We now present a conq~arisonbetween our theoretical

predictions and measured error rates for PS-PSI<
a
the AN/FGC-54, and an PSI< modem, the AN/FGC-29.
modem:
30

Fig. 3. Generic error probability curves for flat fading and atmos-
34

tion of (2B -0.88) decibels [see (117)] is nearly reached a t

10-6 error probability. Since error rates a t HF are due not only to additive noise
When the noises & and 12 are independent, we can repre- but also to selective fading (multipath anddoppler) effects,
sent U in the form some care must be taken to select measured error rates
from experimental data for which multipath and Doppler
#l
J = I p x e 2 r + -1 e2aYe2r (129)
effects contribute very little by comparison with additive
4 4 noise effects. We have selected certain measured valuesob-
where x, are
independent zero mean
unit
variance tained by Korte and Jackson[IG],who in conjunction with
Gaussianvariables.Using (129) in (118) and averaging General Dynanics [17Jhave made extensive comparative
over x, y, we obtain the following double integral repre- tests of the performances of the AN/FGC-29 andthe
sentation for theerror probability, AN/FGC-54, operating with dual diversity.While five sets
of data were collected (designated I-V), on only one of these
(V in [IS]) could it be reasonably inferred that error rate
limitations were due primarily to additive interference.
I n order to use our generic error probability curves we
. dx d3. (130) mustdeterminethe relationship between the measured
This integral was also evaluated with a digital computer SNR and our SNR ( p or 8).J t will be recalled that we de-
for the same set of parameters as in the dependent noise fined p as the ratio of signal-to-noise power inaband-
case and the results are plotted in Fig. 3. Note that the width l / T , where T is the signaling element duration.
maximum reduction in SNR to achieve a given error prob- In theexperimental data the SNR measured was equal to
ability when independent rather than dependent atmos- the ratio of total received signal plus noise power in a
pheric noise occurs is around 1.5 dB aspredicted in (116). centered 2-kc bandwidth of the AN/FGC-29 system to the

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1965 BELLO: ERROR PROBABILITIES IN HF IONOSPEHRIC COMMUNICATIONS 277

I ‘ I ‘ I l l l l l ! l l l l , , , l

Solidlines
denote
theoretical curves.
Dots d e n o tme e a s u r evda l u e s
B I S the pororneter of Lognormal APD which --
approximatesmeasured APD of airnospheric
noiseenvelope. I n HF range, B is approximately
*. equaltoparameter
I
V, measured by NBS.

I I I 1 I 1 , i I 1 1 1 , 1 l , l I
15 20 25
SYSTEM SNR +
Fig. 4. Comparison of measured and predicted error rates for the AN/FGC-29
under flat fading conditions.

noise power in a %kc bandwidth free fromsignal. This Figure 4 compares the measured error probabilities and
same SNR was used for both the AN/FGC-29 and the the theoretical curves for the dependent noise case. Since
,4N/FGC-54.Weshall neglect the fact that signal plus the curvesfordependentandindependent noises differ
noise rather than signal alone was measured, since for the little, we have shown the dependent noise ca’se only in
range of SJXR’s of interest the discrepancy should be small Fig. 4. Only curves with parameters B = 2, 4 are shown,
enough to ignore. since at the frequency of operation the range 2 < B < 4
The AN/FGC-29 modem consists of 16subcarriers covers the most likely situations. The agreenlent between
spaced 170 c/s apart, each modulated with binary FSK a t the theoretical predictions andmeasured error probabilities
a rate of 75 bits per second. The definition of SNR in the is quite good.
experiments yields a value of SNR less than ours, since in TheAN/FGC-54 nlodem consists of 20 subcarriers
theircomputationtheyare,in effect, assigning noise spaced 110 c/s apart each carrying quaternary phase mod-
power for each data channel on the basis of a 170 c/s, ulated data at a rate of 150 bits per second. Although the
bandwidth instead of a 75 c/s bandwidth (1/T = 75 c/s) transmitted signaling elementshave a duration of 1/75
as requiredinour definition. Thus if XFsK denotesour second, the “effective” pulse width is 1/110 = 9.1 111s due
SNR in decibels and s8-sKdenotestheirmeasured SNR to the use of gates in the receiver which gate on only the
in decibels, we have central portion of a pulse. Since the tones are separated
by the reciprocal of the (effective) duration of a signaling
element,their measured SNR should coincide withour
definition for the AN/FGC-54. However, since tjhe same
I n terms of the generic curve F L ( X ) [see (126)] ourpredic- power was transmitted for both the AN/FGC-29 and the
tion of error rate using their definition of SNR is then AN/FGC-54, and since the AN/FGC-54 has 20 tones as
opposed to 16 tones for the AN/FGC-29, their measured
PFGC-29 = FL[SFSK - 2.461. (132) SNR, S F ~ K ,is greater than that which would have been

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278 I E E E TRANSACTIONS ON COMMUNICATION TECHNOLOGY SEPTEMBER

SYSTEM SNR -
Fig. 5 . Comparison of measllred and predicted error rates for the AN/FGC-54
under flat fading conditions.

measured for the AN/FGC-54 system by 10 loglo 20/16 or VI. SUMMARY, AND RECOMMENDATIONS
CONCLUSIONS,
approximately 1 dB. Thus, if SpsK denotes our SNR for Theoretical expressions have been derived for the binary
the AN/FGC-54 system, error probabilitiesof FSK, PT-BPSK, PT-IZPSK, PS-21’sIc,
S P S K = SFSK - 1. (133) and PS-4PSII modems caused by flat fading and atmos-
pheric noise. On the basis of the theory, predictions were
I n terms of the generic curve F ( S ) [see (128)], our pre- made of the binary error probability vs. SNR curves of
diction of the AN/FGC-54 error rate is then the AN/FGC-29 andtheAN/FGC-54 communication
p FGC-54= 2FI,[SFSK - 41. (134) systems. A comparison of these theoretical predictions and
actual measuredperformance shows remarkably good
Figure 5 conlpares the measured error probabilities and agreement.
the theoretical curves for the dependent noise case. Ex- Among theimportant conclusions arisingfrom the
cept for SNR’s less than 12 dB there is good agreement. theory are thefollowing.
We find that higher error rates are predicted below 12 dB 1) In the case of arbitrary noise, the error probability
than were actually measured. However, it hasbeen found8 depends on the noise in thediversity branches only through
that the error probability measurement device could not two generalized noise variables expressible as asum of
measure error rates higher than 2 X lo+ and that when noise powers. pq
such high error rates occurred in a subchnnnel they were 2 ) In the case of PT-PSI< systems in general, and FSI<
generally discarded. This could well account for the error and PS-PSI< systems in the particular case wherein error
rates reported being lower than those predicted since the probability is caused primarily by impulsive noise with a
predicted error rates below 12 dB are near or above 2 X mean frequency muchless than thereciprocal pulse width,
10-2. the error probability depends on a single noise variable

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IO65 ERROR BELLO: PROBABILITIES I N HF IONOSPHERIC COMMUNICATIONS 279

equal to thesum of noise powers at the outputof identical lowed it may be possible to present measured error rate
filters in the various diversity receivers. curves for specific B values and thus reduce the “scatter-
3) As a result of 1) and 2 ) additive noise measurements ing” of measured error probability values noted in Figs.
should be made of the generalized noise variables. 4 and 5 .
4) While in the case of diversity operation, error prob- 9) Since B is the basic parameter for error rate predic-
abilities due to atmospheric noise will be larger than those tions a t HF, it is suggested that measurements of atmos-
due to Gaussian noise at the same SNR; the situation is pheric noise be taken witha view to an accuratedetermina-
reversed in nondiversity operation, Le., lower probabilities tion of this parameter and its st,atistics.
may be expected due to atmospheric noise than to Gaus-
REFERENCES
sian noise. At high SNR’s and nondiversity operation, the
B. Shepelavey, “Non-Gaussianatmospheric noise in binary-
error probability is independent of the noise statistics. data, phase-coherent, communication systems,” ZEEE Trans.
5 ) For any additive noise and Lth order diversity, the on Communication Systems,vol. CS-11, pp. 280-284, September
high SNR error probability decreases as the reciprocal of 1963.
A. D. Spaulding, “Determination of error rates for narrow-
the Lthpower of the SNR and depends upon the additive band communication of binary-coded messages in atmospheric
noise onlythrough the Lth moment of the generalized radio noise,” Proc. IEEE (Correspondence), vol. 52, pp. 220-
221, February 1964.
noise. J. N. Pierce, “Theoretical diversity improvement in frequency-
6) At high SNR and Lthorder diversitythe degradation shift keying,” Proc. IRE, vol. 46, pp. 903-910, May 1958.
P. A. Bello and B. D. Nelin, “The effect of frequency selective
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to Gaussian noise is given by ferentiallycoherentmatched filter receivers,” IEEE Trans.
onCommunicationSystems, vol. CS-11, pp. 170-186, June
1963.
10 (2L - l)! -, “Predetection diversity combining with selectively fading
2B(L - I) - - log,, channels,” IRE Trans. on Cowlwmnication Systems, vol. CS-10,
L LL(L - 1)! pp. 32-42, March 1962.
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E. 0. Sunde, “Digital troposcatter transmission and modula-
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P. A. Be,!lo, Some signal design considerations for HF PSK
between complete dependence and complete independence modems, ADCOM,Inc., Cambridge, _ Mass.,
. Research ReDt.
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“World distribution and characteristics of atmosphericradio
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Methods in Radio Wave
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noise. P. Beckmann,“Amplitudeprobabilitydistribution of atmos-
8) When error rate measurements are made, not only phericradio noise,” RadioSci., vol. 68D, pp. 723-736, June
1964.
should SNR be measured but also the NBS parameter C. W. Helstrom, StatisticalTheory of Signal Detection.
V Dshould be measured. This parameter is equal to the London: Pereamon. 1960.
1 H. Cramer, uMathLmaticalMethods of Statistics. Princeton,
ratio of the rms to the mean of the envelope detected N. J.: Princeton University Press, 1958.
atmospheric noise. The parameter B may be determined 1 J. Korte and C. Jackson, “Evaluation of high-frequency com-
mnnicationsequipment using frequency-stabilized receivers,”
from VD as discussed in the text. If this procedure is fol- U. S. Army Electronics Research and Development Labs., Ft.
Monmouth, N. J., Test Hept. 1544, June 1963.
[17] “Evaluation of new high frequency radio equipment,” General
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Research and Development Labs., Ft. Monmouth, N. J. DA36-039-SC-78309, AS 272565.

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